How can I use the Displace Modifier in a way that allows me to raise an artwork at one spot and recess it on another?
If you really wish to go with displace, and you're working with displacing from an image texture, it should really just be a matter of calibrating the colors of the image texture.

50% gray = no displacement

white = full emboss, displace upwards

black = full deboss, displace downwards

Intermediate shades emboss/deboss a correspondingly intermediate amount

Using GIMP or the like, you should be able to composite your individual displacement maps into a single image so that everything is being displaced in the direction you wish. You may need to invert colors of various parts of the image, or modify the brightness/contrast (e.g., or levels, or curves, whatever) to control the relative amount of displacement.